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We investigate the possibility to discriminate between different pairs of CPnon-conserving mechanisms inducing the neutrinoless double beta$\betabeta$-decay by using data on $\betabeta$-decay half-lives of nuclei withlargely different nuclear matrix elements (NMEs). The mechanisms studied are:light Majorana neutrino exchange, heavy left-handed (LH) and heavy right-handed(RH) Majorana neutrino exchanges, lepton charge non-conserving couplings inSUSY theories with R-parity breaking giving rise to the "dominant gluinoexchange" and the "squark-neutrino" mechanisms. The nuclei considered are$^{76}$Ge, $^{82}$Se, $^{100}$Mo, $^{130}$Te and $^{136}$Xe. Four sets ofnuclear matrix elements (NMEs) of the decays of these five nuclei, derivedwithin the Self-consistent Renormalized Quasiparticle Random PhaseApproximation (SRQRPA), were employed in our analysis. While for each of thefive single mechanisms discussed, the NMEs for $^{76}$Ge, $^{82}$Se, $^{100}$Moand $^{130}$Te differ relatively little, the relative difference between theNMEs of any two nuclei not exceeding 10%, the NMEs for $^{136}Xe$ differsignificantly from those of $^{76}$Ge, $^{82}Se$, $^{100}$Mo and $^{130}$Te,being by a factor $\sim (1.3 - 2.5)$ smaller. This allows, in principle, todraw conclusions about the pair of non-interfering (interfering) mechanismspossibly inducing the $\betabeta$-decay from data on the half-lives of$^{136}Xe$ and of at least one (two) more isotope(s) which can be, e.g., any ofthe four, $^{76}Ge$, $^{82}Se$, $^{100}Mo$ and $^{130}Te$. Depending on thesets of mechanisms considered, the conclusion can be independent of, or candepend on, the NMEs used in the analysis. The implications of the EXO lowerbound on the half-life of $^{136}Xe$ for the problem studied are alsoexploited.

Multiple CP Non-conserving Mechanisms of $\betabeta$-Decay and Nuclei with Largely Different Nuclear Matrix Elements